Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 89 x^{2} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.0521435171923$, $\pm0.947856482808$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}, \sqrt{183})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2121$ | $4498641$ | $10779100164$ | $23777121640041$ | $52599132266695161$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $2032$ | $103824$ | $4872676$ | $229345008$ | $10778984998$ | $506623120464$ | $23811281638468$ | $1119130473102768$ | $52599132297560272$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=6 x^6+22 x^5+40 x^4+20 x^3+41 x^2+10 x+22$
- $y^2=42 x^6+15 x^5+14 x^4+42 x^3+22 x^2+25 x+24$
- $y^2=3 x^6+19 x^5+39 x^4+45 x^3+23 x^2+28 x+13$
- $y^2=6 x^6+2 x^5+37 x^4+18 x^3+28 x^2+35 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47^{2}}$.
Endomorphism algebra over $\F_{47}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}, \sqrt{183})\). |
| The base change of $A$ to $\F_{47^{2}}$ is 1.2209.adl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-915}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.47.a_dl | $4$ | (not in LMFDB) |